Definition:Derivative/Higher Derivatives

From ProofWiki
Jump to: navigation, search


Second Derivative

Let $f$ be a real function which is differentiable on an open interval $I$.

Hence $f'$ is defined on $I$ as the derivative of $f$.

Let $\xi \in I$ be a point in $I$.

Let $f'$ be differentiable at the point $\xi$.

Then the second derivative $f'' \left({\xi}\right)$ is defined as:

$\displaystyle f'' := \lim_{x \mathop \to \xi} \dfrac {f' \left({x}\right) - f' \left({\xi}\right)} {x - \xi}$

Third Derivative

Let $f$ be a real function which is twice differentiable on an open interval $I$.

Let $f''$ denote the second derivate.

Then the third derivative $f'''$ is defined as:

$f''' := \dfrac {\mathrm d} {\mathrm d x} f'' = \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac{\mathrm d^2}{\mathrm d x^2} f}\right)$

Higher Order Derivatives

Higher order derivatives are defined in similar ways:

The $n$th derivative of a function $y = f \left({x}\right)$ is defined as:

$f^{\left({n}\right)} \left({x}\right) = \dfrac {\mathrm d^n y} {\mathrm d x^n} := \begin{cases} \dfrac {\mathrm d} {\mathrm d x} \left({\dfrac {\mathrm d^{n-1}y} {\mathrm d x^{n-1} } }\right) & : n > 0 \\ y & : n = 0 \end{cases}$

assuming appropriate differentiability for a given $f^{\left({n-1}\right)}$.

First Derivative

If derivatives of various orders are being discussed, then what has been described here as the derivative is frequently referred to as the first derivative:

Let $I\subset\R$ be an open interval.

Let $f : I \to \R$ be a real function.

Let $f$ be differentiable on the interval $I$.

Then the derivative of $f$ is the real function $f': I \to \R$ whose value at each point $x \in I$ is the derivative $f' \left({x}\right)$:

$\displaystyle \forall x \in I: f' \left({x}\right) := \lim_{h \mathop \to 0} \frac {f \left({x + h}\right) - f \left({x}\right)} h$

Order of Derivative

The order of a derivative is the number of times it has been differentiated.

For example:

  • a first derivative is of first order, or order $1$
  • a second derivative is of second order, or order $2$

and so on.

Zeroth Derivative

The zeroth derivative of a real function $f$ is defined as $f$ itself:

$f^{\left({0}\right)} := f$

where $f^{\left({n}\right)}$ denotes the $n$th derivative of $f$.